MATEC Web of Conferences 33 , 0 3 0 13 (2015)
DOI: 10.1051/ m atec conf/ 201 5 33 0 3 0 13
C Owned by the authors, published by EDP Sciences, 2015
Model of the Evolution of Deformation Defects and Irreversible Strain at
Thermal Cycling of Stressed TiNi Alloy Specimen
1a
1
1
Aleksandr E.Volkov , Fedor S.Belyaev , Margarita E.Evard , Natalia A.Volkova
1
1
Saint Petersburg State University, 198504 Saint Petersburg, Russia
Abstract. This microstructural model deals with simulation both of the reversible and irreversible deformation of a
shape memory alloy (SMA). The martensitic transformation and the irreversible deformation due to the plastic
accommodation of martensite are considered on the microscopic level. The irreversible deformation is described from
the standpoint of the plastic flow theory. Isotropic hardening and kinematic hardening are taken into account and are
related to the densities of scattered and oriented deformation defects. It is supposed that the phase transformation and
the micro plastic deformation are caused by the generalized thermodynamic forces, which are the derivatives of the
Gibbs’ potential of the two-phase body. In terms of these forces conditions for the phase transformation and for the
micro plastic deformation on the micro level are formulated. The macro deformation of the representative volume of
the polycrystal is calculated by averaging of the micro strains related to the evolution of the martensite Bain’s variants
in each grain comprising this volume. The proposed model allowed simulating the evolution of the reversible and of
the irreversible strains of a stressed SMA specimen under thermal cycles. The results show a good qualitative
agreement with available experimental data. Specifically, it is shown that the model can describe a rather big
irreversible strain in the first thermocycle and its fast decrease with the number of cycles.
1 Introduction
Operational integrity of actuators, sensors, vibration
isolators based on the use of shape memory alloys (SMA)
substantially depends on the irreversible shape change,
which develops at repeated thermocycling of an SMA
part subject to the action of external forces. Thus,
calculation of the irreversible deformation is a challenge
for any theoretical model aimed at the simulation of the
SMA mechanical behavior. A suitable approach can be
microstructural modeling since it is based on the account
of the alloy structure and of the deformation mechanisms.
Probably the first attempt to develop such a model was
the work of Q. Sun and C. Lexcellent [1] who introduced
special internal variables to specify the measures of the
plastic deformation. Similar internal variables were used
in [2, 3]. In these works it was shown that such approach
could describe an incomplete strain recovery on heating.
It also naturally explained the relation between the
unrecovered strain and the two-way shape memory effect.
In works [4, 5] a further step was done. Microplastic
deformation was calculated alongside with the densities
of the scattered and oriented deformation defects, the
isotropic and kinematic hardening being related with
these densities. In the present work a more accurate
account of the oriented defects is done. The density of
these defects is considered to be a state variable,
characterizing the elastic energy of the inter-phase
stresses, rather than the irreversible deformation. The
a
developed model is applied for simulating the variation of
the irreversible deformation on thermocling under
different stresses acting on heating and cooling.
2 Model
This microstructural model uses the approximation of the
small-strain theory. The representative volume is
considered to consist of grains characterized by the
orientation Z of the crystallographic axes. Inside each
grain coexist the austenite and martenside, which in turn
is composed of domains belonging to one of the N
crystallographically equivalent orientations.
2.1 Averaging of microstrains
Reuss’ hypothesis is used for calculation of the strain
tensor H of the representative volume by neutralization of
the strains of grains belonging to this volume:
= 6 ( ),
(1)
where fi and Hgr(Zi) are the volume fraction and the strain
of a grain with the orientation of the crystallographic axes
Zi and the sum is taken over all grains. A grain strain Hgr
is considered as the sum of elastic Hgr e, thermal Hgr T,
phase Hgr Ph and micro plastic Hgr MP components:
= + + + (2)
Corresponding author: author@e-mail.org
This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits XQUHVWULFWHGXVH
distribution, and reproduction in any medium, provided the original work is properly cited.
Article available at http://www.matec-conferences.org or http://dx.doi.org/10.1051/matecconf/20153303013
MATEC Web of Conferences
Also within the Reuss’ hypothesis, we calculate the
elastic strain of a grain by the “mixture rule”:
= (1 ) + (3)
where N is the number of Bain’s orientation variants,
(1/N))n is the volume fraction of the n-th variant of
is the total volume fraction
martensite, = of martensite in a grain. Elastic strains of austenite HA e
and of the n-th variant of martensite HMn e are calculated
by the Hook’s law. Thermal strain Hgr T is calculated in a
similar way with the use of isotropic expansion law for
thermal strains of the two phases.
The phase strain of a martensite variant is the Bain’s
deformation Dn realizing the transformation of the lattice
and (1/N))n is the weight of the n-th variant in the total
phase strain. Thus,
= (4)
Micro plastic strains are caused by the incompatibility of
the phase strains. The main assumption for its calculation
is that the phase strain of a Bain’s variant activates a
combination of slips producing a strain proportional to
the deviator of the phase strain. Thus, for the total micro
plastic strain of a grain one can write:
= N dev
(5)
where internal variables Hnp are measures of the microplastic strains, devDn is the deviator part of tensor Dn, N is
a material constant.
(hereinafter Ms, Mf, As, Af are the characteristic
temperatures and q0 is the latent heat of the
transformation).
As for the elastic inter-phase stress energy Gmix, it
grows with the martensite variants fractions )n and it is
decreased by oriented defects bn, produced by micro
plastic deformations Hnp. In the works [7, 8] the authors
considered martensite variants originated by an invariant
plane deformation and characterized by the habit plane
and the amount of shear. They suggested to calculate the
energy of the variants interaction by a quadratic form. In
the present work the same assumption is used for the
interaction energy of the Bain’s variants of martensite:
= 5, 45 (5 65 )( 6 ),
*
in which the diagonal components of matrix Amn describe
the self-action and the off-diagonal elements – the
interaction of the martensite variants. In TiNi alloy the
basic self-accommodation of martensite is realized by
grouping of the variants into correspondence variants
pairs (CVP) [9-12]. We account for this fact by using the
proper numeration of the variants and taking the matrix
Amn in the form:
A=P 7
(6)
where GA and GMn are the eigen potentials of austenite
and n-th variant of martensite (potentials of the phases as
if they were not interacting), Gmix is the potential of
mixing, which is the elastic inter-phase stress energy. In
formula (6) the eigen potentials are:
= ( ) *
-/0
.- ./0 ,
# ($%$ )'
!"
&
*$&
2 = A, M3,
- ().- (7)
superscript ъ $ stands for austenite and ъ Mn – for the
n-th variant of martensite; T0 is the phase equilibrium
temperature (i.e. such temperature, at which GA = GMn);
G0a and S0a (ъ A, Mn) are the Gibbs’ potentials and the
entropies at stress V=0 and temperature T=T0; Hij0Ta (ъ A,
Mn) are strains of the phases at V=0; cVa and Daijkl (ъ A,
Mn) are the specific heat capacities at constant stress and
the elastic compliances. For T0 we use an estimate
proposed R.J. Salzbrenner, M. Cohen [6]: T0 = (Ms+Af)/2
A1
8;
A1
where the material constant D accounts for the interaction
of the variants in a CVP, and P=q0((Mf–Ms)/T0) / (1–2D).
The thermodynamic force causing growth of the n-th
variant of martensite (i.e. increase of the variable bn) is:
2.2 Transformation conditions
= (1 ) ) + + ,
1 -: -: 0
-:
1 0 -:8
A1 = 7
-: 0 1 -:
0 -: -: 1
A1
> = ?
To formulate the evolution equations for the variables )n
and Hnp we consider generalized forces conjugated with
these parameters. We start from the Gibbs’ potential of
the two-phase grain:
(8)
B
C&
$&
@
@
( ) + D- - E 5 45 (F5 65 )
(9)
There exists a dissipative force opposing the movement
of the phase boundary and responsible for the thermal
hysteresis of the transformation. Denoting this force by
Ffr, we formulate the transformation condition:
Fn = rFfr,
(10)
where the plus sign is for the direct and minus – for the
reverse transformation. The value of Ffr is derived from
the transformation characteristics: Ffr =q0(Ms–T0)/T0.
2.3 Martensite reorientation
A special approach is used to describe the reorientation
(twinning) of martensite. This process is interpreted as a
shift in the space of internal variables )1,…,)N , such
is
that the total amount of martensite = constant. We accept the following hypotheses.
1. Reorientation of martensite in a grain can occur
only if this grain is purely martensitic ()gr = 1). This
hypothesis may seem too restrictive. Still, since
reorientation requires a bigger stress than stress-induced
transformation, it must become the dominating
03013-p.2
ESOMAT 2015
mechanism of deformation only when the direct
transformation is completed.
2. Any variant of martensite can be transformed in
any other variant.
3. Reorientation occurs along the direction in the
space )1,…,)N , which corresponds to the fastest
decrease of the Gibbs’ potential.
4. Reorientation starts when the thermodynamic force
reaches some critical value.
To find the direction of the reorientation we use
HI
HI
vector > = G
,…,
N and take its projection L
HJK
where * is a material constant. First of the equations
(16) reflect the fact that oriented defects grow in a
proportional way with the micro plastic deformation, this
growth being limited by escape to the outer surface. The
second equation (16) means that scattered defects are
proportional to the summary plastic deformation (Odquist
parameter). We assume that the scattered defects give rise
to the isotropic hardening of the material and the oriented
ones – to the kinematic hardening: we relate the defect
densities fn and bn to Fny and FnU by the so called closing
equations, which we choose in the simplest linear form:
HJL
onto plane )1+…+)N =const. Then if for some n it holds
that )n = 0 and Ln<0 we substitute L for its projection Lc
onto intersection of planes )n = 0 and )1+…+)N =const
repeating this procedure for other components of Ln if
necessary. Finally we obtain the unit direction l, which
does not lead to a violation of conditions )n>0, n=1,…,N.
For this direction we postulate the condition of
reorientation:
> OP (Q) = > R OP
(11)
Y
Z
> = 2Y , > = 2Z 6 ,
(17)
where ay and aU are material constants. From conditions
(10) and (14) in the case of the transformation or (11) and
(14) in the case of martensite reorientation using
formulae (9), (12), (13), (15) – (17) evolution equations
relating the increments of the internal variables )n, bn, fn
and to the increments of stress and temperature are
derived. Formulae (1) – (5) allow calculating the
reversible and irreversible macroscopic strain.
where
> OP (Q) = HI
H0
= Q
HI
H)L
= ? Q >
3 Simulation results
(12)
fr tw
In (11) F
is a constant, characterizing the critical force
for reorientation. From hypotheses 2 and 3 it follows that
the increments d)n are proportional to ln :
d)n = ln dM,
(13)
The values of the material constants specifying the
elastic, thermal and phase deformation of SMA were
chosen to reproduce the mechanical behavior of the
equiatomic TiNi SMA. For a specific TiNi composition
experimentally studied in [13] they were determined in
calorimetric and mechanical tests. The values of all
constants are collected in Table 1.
where dM is the proportionality factor to be found from
condition (11).
2.4 Micro plastic flow conditions
Conditions (10) or (11) are insufficient for determination
of the increments of all internal variables. To find the
variation law of variables bn we formulate the microplastic flow conditions:
_Fnp – FnU_ = Fny,
d_Fnp_> 0,
(14)
where Fnp is the generalized force conjugated with the
parameters bn :
HI
> = ?
S
HJL
= E 5 45 (5 6 ),
UV
|6 |HT W(6 HT ),
Material constant
Value
Characteristic
temperatures Mf, Ms,
As, Af
317, 326, 397, 406 K
Latent heat q0
-160 MJ/m3
Number of martensite
variants N
12
Lattice
deformation
matrix D [14]
(15)
Fny and FnU are the forces describing the isotropic and
kinematic hardening. Note that the micro-plastic flow
condition (14) is analogous to the classic plastic flow
condition for 1D case, forces Fnp playing the role of the
stress and Fny, FnU – the roles of the flow stress and the
back stress respectively. Deformation defects generated
by the micro plastic flow we divide in two groups:
oriented defects bn and scattered defects fn, suggesting the
evolution equations for them in the form:
6T = HT Table 1. Values of the material constants.
T = |HT | (16)
0.0188 0.0562 0.0488
[0.0562 0.0188 0.0488 c
0.0488 0.0488 0.0369
Elastic modulus
austenite EA
of
80 GPa
Elastic modulus
martensite EM
of
25 GPa
Poisson’s
ratio
austenite QA
of
0.33
Poisson’s
ratio
martensite QM
of
0.45
Thermal-expansion
03013-p.3
11˜10-6 K-1
MATEC Web of Conferences
b)
coefficient of austenite
Thermal-expansion
coefficient of martensite
6.6˜10-6 K-1
Variants
constant D
0.2
Critical
reorientation
force Ffr tw
20 MJ/m3
Micro plastic
scaling factor N
1.9
strain
irrreversible strain, %
interaction
10
hardening
0.2 MPa
Kinematic
factor aU
hardening
5 MPa
Oriented
defects
saturation factor *
4
2
0
10
20
number of cycle
30
Figure 2. Simulated and experimental [13] dependences of the
irreversible strain increment in one thermocycle under stress
50 MPa (a) and 200 MPa (b) on the cycle number.
2.2
Figures 1 – 4 present the results of simulation of the
strain variation at thermocycling under a constant stress.
7
6
5
strain, %
6
0
Isotropic
factor ay
simulation
experiment
8
4
3
2
The irreversible strain in one cycle decreases with the
number of cycles tending to some small value. The
results of simulation well agree with the experiment for
stress 200 MPa. Not so good agreement for stress 50 MPa
shows that some relaxation process reducing the
deformation hardening are not taken into account.
Active SMA parts in actuators usually experience
thermomechanical cycles in which the stress on heating
Vh is set commonly at bigger values than the stress on
cooling Vc. Naturally, the irreversible strain grows with
the applied stress. An example of the simulation of the
dependence of irreversible strain on the stress Vh is shown
in Figure 3.
0
300
350
400 450 500
temperature, K
irrreversible strain, %
1
550
Figure 1. Simulated strain variation at thermocycling under a
constant stress 50 MPa.
4
2
0
a)
-200 -100
0 100 200 300 400
stress at heating, MPa
Figure 3. Dependence of the irreversible strain in the first
thermocycle on the stress applied at heating. Stress at cooling is
50 MPa (simulation).
Note that the irreversible strain grows with Vh faster
after it has exceeded some critical value. An analysis
shows that this acceleration is related to the activation of
the martensite reorientation when the force needed for it
reaches the critical value.
Figure 4 presents the strain variation in the course of
three thermocycles with Vc = 50 MPa, Vh = 200 MPa.
03013-p.4
ESOMAT 2015
Simulation predicts that the total irreversible strain
growth must be influenced stronger by the stress applied
on cooling than on heating.
12
strain, %
10
8
Acknowledgements
6
This research was supported by the grant of Russian
Foundation of Basic Research 15-01-07657.
One of the authors (A.E. Volkov) acknowledges SaintPetersburg State University (Russia) support of
presenting this work at the conference ESOMAT-2015
(project 6.41.647.2015).
4
2
0
300
350
400 450 500
temperature, K
550
References
One can see a typical “hump” on the strain dependence
on temperature while heating. It is related to the
reorientation of martensite occurring along the sequence
“martensite – virtual austenite – martensite of other
orientation” prior to the start of the reverse
transformation.
Figure 5 shows the growth of the total irreversible
strain on thermocycing. The irreversible strain after 30
cycles under Vc=50 MPa, Vh= 200 MPa is only two times
larger than after 30 cycles under Vc=Vh= 50 MPa. At the
same time cycling under Vc=Vh= 200 MPa results in a
much bigger growth of the irreversible strain.
2.
accumulated irrreversible strain, %
Figure 4. Strain variation under thermocycling at stress on
cooling Vc=50 MPa and stress on heating Vh= 200 MPa
(simulation).
1.
3.
4.
5.
6.
7.
40
8.
30
Vc = 200 MPa, Vh = 200 MPa
Vc = 50 MPa, Vh = 50 MPa
Vc = 50 MPa, Vh = 200 MPa
20
9.
10.
10
0
11.
0
10
20
number of cycle
12.
30
Figure 5. Dependence of the total accumulated irreversible
strain on the number of thermocycles carried out under different
stresses Vc applied on cooling and Vh applied on heating
(simulation).
13.
14.
Summary
Incorporation of the deformation defects conception into
a microstructural model opens the possibility of
simulating the irreversible strain acquired by an SMA
specimen at thermomechanical cyclic loading under
different stresses applied on the stages of cooling and
heating. An account of the deformation defects
production and of their effect on the isotropic and
kinematic hardening allows describing the decrease of the
irreversible strain in one cycle with the cycle number.
This data is important for designing of SMA actuators.
03013-p.5
Q.-P. Sun, C. Lexcellent, J. de Physique IV. 6
(1996), C1-367-375.
M.E. Evard 121, A.E. Volkov, J. Eng. Mater. and
Technol. (1999), 102–104.
A.E. Volkov,
F. Casciati,
in:
F.Auricchio,
L.Faravelli, G.Magonette and V.Torra (Eds.) Shape
Memory Alloys. Advances in Modelling and
Applications, Barcelona, 88-104 (2001).
F.S. Belyaev,
M.E.Evard,
A.E. Volkov,
N.A. Volkova, Materials Today Proceedings. (to be
published)
A.E. Volkov, M.E. Evard, F.S. Belyaev, Materials
Science Foundations, 81-82 (2015), 20-37.
R.J. Salzbrenner, M. Cohen, Acta Met., 27, N 5
(1979), 739–748.
N. Siredey, E. Patoor, M. Berveiller, A. Eberhardt,
Int. J. of Solids and Structures 36 (1999), 4289-4315.
Y. Chemisky, A. Duval, E. Patoor, T. Ben Zineb,
Mechanics of Materials 43 (2011) 361–376.
K. Madangopal, J. Singh, Benerjee, Scr. Met., 25,
2153–2158 (1991).
K. Madangopal, J. Singh, Acta Mater., 48 (2000),
1325-1344.
M. Nishida, T. Nishiura, H. Kawano, T. Imamura,
Phil. Mag., 92 (2012), 2215–2233.
M. Nishida, E. Okunishi, T. Nishiura, H. Kawano,
T. Imamura, S. Ii, T. Hara, Phil. Mag., 92 (2012),
2234–2246.K.M. Knowles, D.A. Smith, Acta Met.,
20, 101–110 (1981).
S. Belyaev, N. Resnina, A. Sibirev, J. Mater. Eng.
Perform., 23 2339–2342 (2014).
T. Imamura, T. Nishiura, H. Kawano, H. Hosoda,
M. Nishida, Phil. Mag., 92 (2012), 2247–2263.